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April 5, 2011

Homotopy Type Theory, IV

Posted by Mike Shulman

So far in this series we’ve described the correspondence between type theory and homotopy theory and defined some basic notions of homotopy theory in type theory, including equivalences in several ways. We’ve also mentioned a few axioms that we may want to add to intensional type theory, including “function extensionality”, a subobject classifier, and a truncation π−1=τ≤−1\pi_{-1} = \tau_{\le -1} into (-1)-types.

However, nothing we’ve said so far excludes the possibility that all types are discrete (= 0-types = sets). Intensional type theory plus function extensionality has sound semantics in any locally cartesian closed 1-category; and if the category is regular, then τ≤−1\tau_{\le -1} exists; while if it is a topos, then of course it has a subobject classifier. But today I’ll introduce Voevodsky’s univalence axiom, which is not valid in any 1-categorical semantics—or, indeed, in nn-categorical semantics for any finite nn! The univalence axiom is perhaps the easiest and most intuitive way to require that our homotopy type theory is honestly “homotopical”, and it also has other pleasant consquences (including, perhaps surprisingly, function extensionality).

In Voevodsky’s original phrasing, the univalence axiom is an augmentation of universes, which are a type-theoretic notion that I haven’t mentioned yet. (If you don’t like universes, that’s okay; carry on reading.) At an informal level, I think type-theoretic universes are not very different from the universes of the Grothendieck sort that you may be more familiar with, and are even more closely related to their categorial analogues. In terms of categorical semantics, a universe is a type UU, together with a display map E→UE\to U (that is, a type E(u)E(u) dependent on u∈Uu\in U). We think of the elements of UU as “codes for types”, with u∈Uu\in U coding for the type E(u)E(u). And we require that the type-theoretic operations, such as dependent sums and products, are represented by operations on UU, so that types of the form E(u)E(u) are closed under such operations.

Type-theoretically, we usually identify types E(u)E(u) with their codes u∈Uu\in U, so that the elements (terms) of UUare types. We generally assume every type is contained in some universe, so that we can replace judgments of the form “AA is a type” with “A∈UA\in U” for some universe UU. In particular, any universe U=U0U=U_0 must be an element of some universe U1U_1, which must be an element of some universe U2U_2, and so on; we often postulate that every type belongs to one of a specific sequence of universes U0∈U1∈U2∈…U_0 \in U_1\in U_2\in\dots. Frequently a universe is written as “TypeType” or “TypenType_n”.

Thus a universe is a “type of types.” If we regard types as sets, then this is like a set of sets. But if we are category theorists, we know that it’s unnatural to have a set of sets; really we should have a category, or at least a groupoid, of sets. And we should have a 2-groupoid of groupoids, and an (n+1)(n+1)-groupoid of nn-groupoids, and so on. But the nice thing about ∞\infty is that ∞=∞+1\infty = \infty+1, so that we can expect to have an ∞\infty-groupoid of ∞\infty-groupoids. Thus, arguably, it’s really in the homotopy context that the notion of universe is “most sensible”.

Now it’s all well and good to say we have an ∞\infty-groupoid of ∞\infty-groupoids, but what is that ∞\infty-groupoid? Its objects are of course ∞\infty-groupoids, but we also know what its morphisms should be, and its 2-morphisms, and so on: they should be the equivalences, homotopies, and so on between ∞\infty-groupoids. However, the basic type-theoretic notion of universe doesn’t tell us anything about what the path-types of the universe are like; this is what the univalence axiom fixes. (It’s analogous to how plain intensional type theory doesn’t tell us anything about when two functions should be considered equal; hence we need function extensionality.)

To make things more precise, let AA and BB be types in some universe UU; we want to specify what PathsU(A,B)Paths_U(A,B) should be. And we have a natural candidate, namely the type

of equivalences from AA to BB. (Remember that IsEquiv(f)IsEquiv(f) is a proposition, so it makes sense to think of points of Equiv(A,B)Equiv(A,B) as functions A→BA\to B with the property of “being an equivalence.”) Moreover, we have a natural map

How do we define externU,A,Bextern_{U,A,B}? Remember from the first post that the “elimination rule” for path-types says:

Given a type C(x,y,p)C(x,y,p) which may depend on two points x,y∈Xx,y\in X and a path p∈PathsX(x,y)p\in Paths_X(x,y) between them, if we have a way to produce an element of C(x,x,rx)C(x,x,r_x) for any x∈Xx\in X, then we can “transport” it along any path p∈PathX(x,y)p\in Path_X(x,y) to produce a canonical element of C(x,y,p)C(x,y,p) (and in such a way that if we transport it along rxr_x then it doesn’t change).

We’re going to apply this rule with X=UX=U, x=Ax=A, and y=By=B. We’ll take the type C(A,B,p)C(A,B,p) to be Equiv(A,B)Equiv(A,B), which depends on AA and BB and (vacuously) a path between them. Now we do have a way to produce an element of C(A,A,rA)=Equiv(A,A)C(A,A,r_A) = Equiv(A,A), namely the identity function 1A:A→A1_A\colon A\to A (which is an equivalence; I’ll leave proving that as an exercise). Therefore, we can transport the identity 1A1_A along any path p∈PathsU(A,B)p\in Paths_U(A,B) to produce a canonical element of Equiv(A,B)Equiv(A,B) corresponding to pp. This defines the map externU,A,Bextern_{U,A,B}.

Let’s think first about the semantics of univalence. First of all, in the form I stated it above, it is an axiom about a particular universe UU. A universe satisfying the univalence axiom is called a univalent universe. We generally assume that all of the specified universes U0∈U1∈U2∈…U_0 \in U_1\in U_2\in\dots are univalent.

In the “standard” model in ∞\infty-groupoids, we obtain a univalent universe from “the ∞\infty-groupoid of all ∞\infty-groupoids bounded in size by some inaccessible cardinal κ\kappa”. Thus, if there are arbitrarily large inaccessibles, every type will belong to some univalent universe. (I’m not sure whether inaccessibles are necessary here or whether some weaker assumption would suffice.) I believe this is the only model with enough univalent universes that has been constructed in set theory with anything approaching rigor (by Voevodsky).

However, I think most people expect that in more general (∞,1)(\infty,1)-categorical semantics, we ought to obtain a univalent universe from any object classifier with strong enough closure properties. In particular, in any (Grothendieck) (∞,1)(\infty,1)-topos, there ought to be a univalent universe of all “κ\kappa-compact” types, for any inaccessible κ\kappa.

Moreover, any “full subuniverse” of a univalent universe will again be a univalent universe, as long as it is closed under the type-theoretic operations. In particular, if UU is any univalent universe, then its full subuniverse of nn-types is again univalent for any nn, and that subuniverse will itself be an (n+1)(n+1)-type. Thus a univalent universe need not itself be of infinite h-level: we can have a univalent groupoid (1-type) of small sets (0-types), a univalent 2-groupoid (2-type) of small groupoids, and so on. At the bottom, we can have a univalent set (0-type) of small truth values ((-1)-types).

In particular, a subobject classifier, if one exists, is also a univalent universe; so to get ourselves out of the world of sets we need at least two univalent universes. Similarly, we can have a sequence of univalent universes U0∈U1∈U2∈…U_0 \in U_1\in U_2\in\dots in which U0U_0 contains only (-1)-types and is itself a 0-type, U1U_1 contains only 0-types and is itself a 1-type, and so on. Such a stratification of universes by “categorical dimension” as well as by size does seem to match much of mathematical practice—but only outside of homotopy theory. For homotopy theory, we do really want to have ∞\infty-types that aren’t nn-types for any finite nn (such as, for instance, the 2-sphere S2S^2), and an infinite sequence of univalent universes doesn’t seem to be enough to guarantee this. I’ll come back to this later.

Vladimir explained the origin of the word “univalent” as follows:

a universal fibration is one of which every other fibration is a pullback in a unique way (up to homotopy).

a versal fibration is one of which every other fibration is a pullback in some way, not necessarily unique.

a univalent fibration is one of which every other fibration is a pullback in at most one way (up to homotopy).

Thus the univalence axiom asserts that the structural fibration of the universe is univalent.

Now, the principal way we use the univalence axiom is as follows: given an equivalence f:A→Bf\colon A \to B, we apply the inverse of externU,A,Bextern_{U,A,B} to get a path f^∈PathsU(A,B)\hat{f}\in Paths_U(A,B), then apply the above-mentioned “elimination rule” for elements of path-types. Putting this together, we get the following consequence of univalence, apparently first formulated by Peter Lumsdaine and Andrej Bauer.

Given a type C(A,B,f)C(A,B,f) which may depend on two types A,BA,B and an equivalence f:A→Bf\colon A\to B between them, if we have a way to produce an element of C(A,A,1x)C(A,A,1_x) for any type AA, then we can “transport” it along any equivalence f:A→Bf\colon A\to B to produce a canonical element of C(A,B,f)C(A,B,f) (and in such a way that if we transport it along 1A1_A then it doesn’t change).

The elimination rule for paths is sometimes called path induction, since it is an instance of the general induction principle for inductively defined types. By analogy, we refer to the above consequence of univalence as equivalence induction. Informally, it means that

Given an equivalence f:A→Bf\colon A\to B, we can “identify” BB with AA along ff. Specifically, in any construction we can perform, or theorem we can prove, starting only from a type AA, we can obtain another valid construction or theorem by replacing some copies of AA with BB and any necessary occurrences of 1A1_A by ff. Behind the scenes, this replacement uses ff and its inverse to silently transfer data back and forth between AA and BB as necessary.

Such “identification” of course a very common thing to do in mathematics, often without even remarking on it! But usually, if it is justified at all, it is “by abuse of notation” or by trusting the reader to do the translation. The univalence axiom formalizes it, makes it happen “automatically” in the background, and makes it “natural/continuous.”

Moreover, equivalence induction actually implies the full univalence axiom. For if we apply equivalence induction to the type C(A,B,f)≔PathsU(A,B)C(A,B,f) \coloneqq Paths_U(A,B) and the identity path rA∈PathsU(A,A)r_A\in Paths_U(A,A), we obtain a way to make any equivalence f:A→Bf\colon A\to B into a path f^∈PathsU(A,B)\hat{f}\in Paths_U(A,B). The final condition that transporting along the identity equivalence leaves something unchanged (together with the same property for the identity path) then makes this construction into an inverse of externU,A,Bextern_{U,A,B}. I’ve checked this in Coq. But it’s not really surprising, because equivalence induction gives the type Equiv(A,B)Equiv(A,B) the “same inductive/universal property” as PathsU(A,B)Paths_U(A,B). (But I don’t know how to state equivalence induction in a way that is evidently a proposition.)

Note that equivalence induction makes no reference to a particular universe containing AA and BB, except that the type C(A,B,f)C(A,B,f) is required to be defined “parametrically” for all A,BA,B in the universe. In particular, this implies that if U1U_1 is a univalent universe and U2U_2 is a “larger” universe, in the sense that every type in U1U_1 also belongs to U2U_2, then externU2,A,Bextern_{U_2,A,B} is also an equivalence for any A,B∈U1A,B\in U_1, whether or not U2U_2 itself is univalent. (It can apparently still be the case that a “smaller” non-univalent universe is contained in a “larger” univalent one, however.) So univalence is almost a property of types (or pairs of types) rather than a property of universes. Furthermore, we can make sense of equivalence induction even if there are no universes, if instead we have some sort of “polymorphism” allowing us to make sense of “defining a type parametrically over other types”. (Thanks to Peter Lumsdaine for correcting some errors in the original version of this paragraph; see his comment and ensuing discussion below.)

Univalence also implies other useful things, like function extensionality and (maybe, with some help) quotients, but let’s save those for another day.

Posted at April 5, 2011 5:30 AM UTC

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17 Comments & 2 Trackbacks

Re: Homotopy Type Theory, IV

another great posting, Mike!
I especially appreciate the new observation:

“… So contrary to its original appearance, univalence can be considered to be a property of types (or, perhaps, pairs of types) rather than a property of universes. Furthermore, we can make sense of equivalence induction even if there are no universes …”

As you say, understanding univalence in terms of the associated “equivalence induction principle” corresponds to the usual mathematical practice of “identifying” equivalent structures – and makes it rigorous rather than just careful sloppiness.

It’s (i) admissible by a property of type theory, namely being “homotopy invariant” in the sense that anything expressible/definable/provable is stable under a transformation along an equivalence (and this is *proved* by VV’s model in Kan complexes, which shows that adding UA is formally sound); and (ii) it expresses a commitment not to add any new constructions, terms, axioms, that would break that property (not that anyone would contemplate doing such a thing).

Already at the level of 1-types, it has a pleasant consonance with mathematical practice not shared by conventional foundations, since it allows us to treat isomorphic structures as “identical” – something that has seemed puzzling from the point of view of set-theoretic foundations, since the language of set-theory doesn’t have the invariance property (i).

BTW: you didn’t mention how to actually get a univalent universe from “the ∞\infty-groupoid of all ∞\infty-groupoids (bounded in size)”. Vladimir’s construction involves the theory of minimal fibrations, well-ordering of the fibers, and other technology. Andre Joyal suggested an alternate construction at Oberwolfach – perhaps Nicola Gambino would be willing to post it to the HoTT site?

Re: Homotopy Type Theory, IV

it has a pleasant consonance with mathematical practice not shared by conventional foundations, since it allows us to treat isomorphic structures as “identical”

Yes, that’s a nice point; the univalence axiom takes the idea of categorical/structural set theory one step further. In structural set theory, we can’t distinguish between isomorphic structures (your invariance property (i)), but we still have to explicitly insert isomorphisms to pass between them. (For other readers: I didn’t mention the fact that univalence, which is directly about equivalences between types, also implies a corresponding fact for structured types, but it’s true.)

Univalent type theory takes the lesson from structural set theory (and, arguably, most of 20th century mathematics) that in general, we need to remember “an isomorphism” rather than the mere fact of “being isomorphic”. But then it says that we can treat isomorphisms almost like equalities, coming even closer to the structuralist ideal.

Let me take this opportunity to mention, for new readers, another philosophical/terminological issue, of which I was reminded by the fact that you put “identical” in quotes. Where I come from, it’s important to distinguish between equality and isomorphism/equivalence. And while the univalence axiom collapses Martin-Lof’s “identity types” with “equivalences,” it seems to me as though it does it by making the “identity types” behave like spaces of equivalences, rather than the reverse. So I prefer to use the term “path types” and say that “univalence reinterprets the path types of universes to refer to equivalence rather than to equality”. However, a number of people (particularly type theorists) seem to like to say instead that “univalence makes equivalent structures equal”—and I guess that if you come from a world where the behavior of intensional identity types is what you are used to thinking of as “equality”, that makes sense. I don’t know whether it’ll ever sound right to me, though. (-:

you didn’t mention how to actually get a univalent universe

That’s right; because I haven’t understood the details! I’d love to see it explained, and I’d especially like to free it from things like minimal fibrations, which I expect would be necessary in order to extend it to object-classifiers in more general categories.

Re: Homotopy Type Theory, IV

Very interesting suggestion that equivalence-induction allows one to consider univalence without universes! Morally I agree something like that should hopefully be the case, but I’m not quite convinced it’s true as stated, unless I’m missing something.

The problem is that the “eliminator” for equivalence-induction still does refer to a universe U: the type C(A,B,f) one eliminates into must vary over types A, B : U, and equivalences between them. Even though we’re not explicitly referring to paths in U, U appears in the dependency of the target type, and so its path spaces are implicitly involved.

In particular, I think one can have types A, B which belong to both a univalent universe U₁ and a non-univalent one U₂. As we discussed at one point, one should be able to get this within the groupoid model, given a notion of “small” (eg <κ): take U₁ to be the groupoid of small groupoids, and U₂ to be the discrete groupoid on the set of small groupoids. So here, extern(U₂,A,B) may not be an equivalence even if extern(U₁,A,B) is?

I guess the minimum one needs to state equivalence-induction is something like (a) the ability to have types dependent over type-variables, i.e. some form of polymorphism, whether using a universe or otherwise; and (b) a type Equiv(A,B), for any types A, B?

Re: Homotopy Type Theory, IV

Thanks for keeping me honest! It seems to me that you raise two separate issues: (1) what we need from our type theory in order to state equivalence induction without a universe, and (2) if we do have a univalent universe, whether the types in that universe can also belong to another universe and fail to be univalent there.

Re: (1), I agree that the issue needs to be addressed. But I’m not convinced that we need something beyond plain universe-free dependent type theory, as long as the latter includes judgments of the form “AA is a type” — which it seems as though it had better do, as soon as we have any type constructors. Can’t we then state equivalence-induction as a rule like this?

Re: (2), you’re right that I got overenthusiastic. I think what I should have said was “if U1U_1 is any univalent universe, and U2U_2 is a larger universe than U1U_1, in the sense that all types in U1U_1 are also in U2U_2, then externU2,A,Bextern_{U_2,A,B} is also an equivalence for any A,B∈U1A,B\in U_1.” The point being that in order to apply equivalence induction, we need the type PathsU2(A,B)Paths_{U_2}(A,B) to be defined parametrically for all A,B∈U1A,B\in U_1, hence we need all types in U1U_1 to be in U2U_2.

Semantically, this means we have to have a map U1→U2U_1\to U_2, which is lacking in your example. So although it is true, externally, in that example, that “every type in U1U_1 is also in U2U_2,” it’s not true type-theoretically, i.e. we don’t have an inference rule like A:U1⊢A:U2A\colon U_1 \vdash A\colon U_2. Does that seem right?

Re: Homotopy Type Theory, IV

Re (1), we need afaics (as your rule shows) not just judgements “A type” — which one has in all variants of M-L Type Theory — but type variables that can occur in contexts, which one doesn’t normally have in M-L TT, except via universes.

Allowing type variables as a primitive notion — not via a universe — I think of as being the key idea of polymorphism; but I’m not at all familiar with polymorphic type theories, so I don’t know how much what one could do with this in that setting…

Re: Homotopy Type Theory, IV

Polymorphic type theory is wonderful. At the small price of trading powersets for polymorphic indexing, everything you could possibly want to work, works.

By “everything”, I mean that polymorphic types form a small complete category, which is not degenerate. Furthermore, polymorphic types have a natural external notion of equality via parametricity, which works ridiculously well. Unfortunately, internalizing parametric equality into polymorphic type theory has been a very challenging problem.

I think that homotopy-style ideas would help in making it work. The reason for this hunch is that the usual way of constructing models of polymorphic types is as partial equivalence relations on some universal domain, and I suspect a move to groupoids would be helpful in figuring out what to do.

Re: Homotopy Type Theory, IV

Re (2), I think I agree. In a syntax where the “elements” fibration EE is explicit, I guess we want to assume rules something like
x:U1⊢coerce(x):U2x:U1⊢E2(coerce(x))=E1(x)type
x \colon U_1 \vdash coerce(x) \colon U_2
\qquad
x \colon U_1 \vdash E_2(coerce(x)) = E_1(x) type
and then I think I agree with your original statement.

Re: Homotopy Type Theory, IV

Allowing type variables as a primitive notion — not via a universe — I think of as being the key idea of polymorphism

That makes sense, I guess. So I guess you are right that “some form of polymorphism” is necessary in order to state equivalence-induction. But I think it is a fairly weak form of polymorphism, in that we don’t need to be able to quantify over type-variables, we just need to allow them in contexts. Semantically, I think one ought to be able to make sense of it in any categorical model.

Re: Homotopy Type Theory, IV

I’ve edited the post slightly, in an attempt to avoid misleading new readers. Thanks again!

I almost feel now as though the polymorphic approach would be better than the one that uses universes. Although I suppose that if you have polymorphism and universes, then the polymorphism form of equivalence-induction wouldn’t imply univalence for the universes, since the path-types of a universe are not defined parametrically for all types, only for those types belonging to the universe. I think there’s something going on there that I don’t understand yet.

Perspective

Yesterday Bas Spitters was so kind to give me a demonstration of Coq in action, learning and then doing proofs in homotopy theory. Seeing this is quite thrilling. As if a character of a novel one has been reading suddenly walks into the room and stands in front of you, fully embodied.

While I understand that the HoTT project is only just getting off the ground and needs more work, I’d be interested in getting a better idea of what one can reasonably expect to be eventually possible here.

First of all I gather that the immediate next major goal is to refine the axioms such as to give a complete axiomatization of ∞\infty-toposes. I suppose. I am not sure how explicitly and widely this goal has been stated. Seems to me to be the central goal to go for. Bas points out that Steve Awodey mentions it at the very end of his notes Type theory and homotopy .

What then? Is it conceivable that one can add further properties to the ∞\infty-toposes thus axiomatized? Can we expect to “have a homotopy type theory of locally ∞\infty-connected ∞\infty-toposes”? Or some such statement?

You can probably guess what I am getting at: since I am claiming that plenty of differential geometry/differential cohomology theory can be axiomatized formally in any cohesive ∞\infty-topos, I am wondering if the axioms of cohesive ∞\infty-toposes in turn can be founded entirely formally in type theory. That would seem be a striking consequence of the striking fact that their axioms are so simple! (We can ignore the extra axioms “pieces have points” and “discrete objects are concrete axioms” for the moment as just extra icing. The core axiom is just the extra left and right adjoint to the terminal geometric morphism. That alone supports the core theory.)

If so, that would seem to immediately give a foundation of structures like Chern-Weil theory and Chern-Simons theory (and their higher generalization) in type theory, given that I am claiming that both are axiomatized in any cohesive ∞\infty-topos.

(Maybe I am not saying the bits about “foundations” and “axiomatized” in the right words, hope you can nevertheless see what I have in mind).

Re: Perspective

Could this be related to Louis Crane’s work on model categories and physics?

I do not see indication that it would. Also I don’t understand what the article you point to has to do with model categories (if that is what you were thinking of), except that the word appears at one point. Do you?

Maybe let’s not get into physics here. What I said above lives in pure math: the claim is that in an ∞\infty-sheaf ∞\infty-topos that is equipped with an extra left adjoint and an extra right adjoint to its terminal geometric morphism, such that the extra left adjoint preserves finite products, the following notions, for instance, have simple axiomatization: de Rham cohomology, ordinary differential cohomology, Chern-Weil theory, Chern-Simons functionals. (This is discussed here).

This is in the sense that: there are simple constructions on objects in such an ∞\infty-topos just involving ∞\infty-(co)limits and the four adjoint ∞\infty-functors assumed above, such that 1. these constructions have the general abstract properties of the notions named above, and 2. such that when realized in a specific model of the axioms (discussed here) these reproduce the traditional notions that go by these names.

While working on this, I was struck by how much indeed does follow from just the assumption of an ∞\infty-topos with such two extra adjoints functors. Originally Lawvere had suggested that this is so for ordinary toposes. But it turns out that the same simple axiom put on an ∞\infty-topos implies a pretty comprehensive supply of advanced differential-geometric notions.

To me, this seems to provide a foundation of these “advanced differential-geometric notions” quite deeply inside ∞\infty-topos theory. Notably all the notions can be expressed with nothing else but the assumption that there is an ∞\infty-topos with these two extra adjoint functors.

So if there were a homotopy-type-theoretic way to axiomatize not just “∞\infty-topos” but “cohesive ∞\infty-topos” that would seem to imply in turn a homotopy-type-theoretic way to axiomatize rather more complex-sounding notions such as “Chern-Weil theory”.

Therefore my question: is it conceivable that HoTT can eventually axiomatize not just the notion “∞\infty-topos” (as people seem to expect) but moreover notions of ∞\infty-toposes with extra structure and properties, such as “∞\infty-topos that is local, locally ∞\infty-connected and ∞\infty-connected over another ∞\infty-topos”?

Re: Perspective

It seems that I used the wrong link. This is Crane’s work on Model Categories and Quantum Gravity. I merely wanted to suggest the possibility that this may be another possible application of the internal language, not that is related to the application that you mention. The paper seems a bit informal, so it is hard to judge what is going on precisely.

Re: Perspective

Hi Bas,

thanks for the link, now I see what you mean. I could comment on the content of the link you give, but now it occurs to me that you just meant to point to some possible occurence of homotopy theory in physics. Is that right?

In that case I’d stress that there are plenty of well established and well-studied occurences of homotopy theory in physics (fundamental physics, that is), hence of model categories and of ∞\infty-categories and ∞\infty-toposes, albeit the latter are of course made explicit by fewer authors than use them implicitly. Some such occurences are listed here.

But I thought from the point of homotopy type theory what would be interesting are situations where not only some homotopy theoretic concepts appear, but where a situation can entirely be formally axiomatized using the internal language of a certain class of ∞\infty-toposes. For in combination this would seem to in principle make such situations accessible to computer-aided proof.

So to come back to my questions, starting with the simplest one:

it seems that there should be a way to massage the present axioms of HoTT such as to give an axiomatization of the notion of ∞\infty-toposes. Is it conceivable that it is possible to massage the axioms further such as to encode ∞\infty-toposes over another ∞\infty-topos?